The spectral decomposition of shifted convolution sums

Let pi(1), pi(2)) be cuspidal automorphic representations of PGL(2)(R) Qf conductor 1 and Hecke eigenvalues lambda(pi 1,2) (n) and let h > 0 be an integer. For any smooth compactly supported weight functions W-1,W-2 : R-x --> C and any Y > 0, aspectral decomposition of the shifted convolution sum Sigma(m+/-n=h) lambda(pi 1) (vertical bar m vertical bar)lambda(pi 2) (vertical bar n vertical bar)/root vertical bar mn vertical bar W-1(m/Y)W-2(n/Y) is obtained. As an application, aspectral decomposition of the Dirichlet series [GRAPHICS] is proved for Rs > 1/2 with polynomial growth on vertical lines in the S-aspect and uniformity in the h-aspect.